Suppose that 65% of all registered voters in a certain area favor a 7-day waiting period before purchase of a handgun. Among 285 randomly selected voters, determine the approximate probabilities below. (Use the approximating normal distribution. You may need to use a table. Round your answers to four decimal places.) USE SALT (a) What is the approximate probability that at least 190 favor such a waiting period? 0.2901✔ (b) What is the approximate probability that more than 190 favor such a waiting period? 0.2796✔ (c) What is the approximate probability that fewer than 170 favor such a waiting period? 0.0332 X
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Determine the Parameters of the Binomial Distribution: Given that 65% of voters favor the waiting period, the parameters of the binomial distribution are:
Probability of success (p) = 0.65- Number of trials (n) = 285
- Probability of failure (q) = 1 - p = 0.35
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Compute the Mean and Standard Deviation: For the binomial distribution:
- Mean (μ) = n * p = 285 * 0.65 = 185.25
- Standard Deviation (σ) = sqrt(n * p * q) = sqrt(285 * 0.65 * 0.35) ≈ 8.439
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Calculate the Z-score: For the question of fewer than 170 people favoring, the target is X = 170. We use continuity correction by subtracting 0.5 to get 169.5. The Z-score formula is: z=X−μσz=σX−μ z=169.5−185.258.439z=8.439169.5−185.25 This results in a z-value of approximately -1.866.
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Use the Z-score to Determine the Probability: A Z-table or standard
normal distribution table is used to find the probability associated with a given z-value. For z = -1.866, the probability is approximately 0.0311.
Therefore, using the steps outlined, we can conclude that the probability that fewer than 170 voters favor the waiting period is approximately 0.0311. However, it's important to note that while the method is standard, rounding or different z-tables might give slightly varying results.
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